Bulk acoustic wave accelerometers

ABSTRACT

Accelerometers and associated techniques for detecting motion are described. For a resonant accelerometer, an externally-applied acceleration can cause a change in the electrical spring constant K c  of the electromechanical system. A resonant accelerometer can be driven to resonate in a bulk acoustic wave mode of vibration, which can have a high resonant frequency. Other accelerometers and associated techniques are disclosed.

RELATED APPLICATION

This application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Application Ser. No. 60/979,131, entitled “MULTI-AXISCAPACITIVE BULK ACOUSTIC WAVE ACCELEROMETERS,” filed on Oct. 11, 2007,which is herein incorporated by reference in its entirety.

DISCUSSION OF RELATED ART

Accelerometers are devices that can be used to sense or measureacceleration along or around one or more axes. Some applications ofaccelerometers include sensors for automobile air bag deployment, andportable electronic devices such as video game controllers.Accelerometers that are small in size are known as microaccelerometers.At least two types of microaccelerometers exist in the market spacetoday that use different techniques for measuring external acceleration.

One class of conventional microaccelerometers sense acceleration bymeasuring changes in the displacement of a proof mass caused by anexternally-applied acceleration. This displacement is typically measuredcapacitively by detecting a change in the size of the capacitive gaps ora change in the amount of capacitive overlap area, when the proof massis displaced by an externally applied acceleration. Accelerometers inthis class are known as static accelerometers.

Another class of accelerometers, known as resonant accelerometers, senseacceleration by exciting a micromechanical structure into its resonantfrequency. The accelerometer natural angular frequency may be expressedby ω₀=√{square root over (K/M)}, where M is the proof mass of theaccelerometer, and K is the effective spring constant of theaccelerometer. The accelerometers are typically designed such thatK_(electrical)<<K_(mechanical). When an external acceleration isapplied, the mechanical stiffness or mechanical spring constant(K_(mechanical)) changes as a result of the structural strain imposed bythe external acceleration. Acceleration can be detected based on achange in the resonant frequency that is caused by the change in themechanical spring constant K_(mechanical). Accelerometers in this classare known as resonant accelerometers.

SUMMARY

Some embodiments relate to resonant accelerometers having anelectromechanical system that resonates in a bulk acoustic wave mode ofvibration. A bulk acoustic wave mode of vibration may have a resonantfrequency on the order of 1 MHz to 50 MHz, for example. Anacceleration-induced change in the electrical spring constantK_(electrical) or a related parameter may be used to measureacceleration. In some embodiments, when an external acceleration isapplied, the resonant frequency of the system may change in response toa change in the electrical spring constant K_(electrical). The change inthe electrical spring constant K_(electrical) may result from a changein the size of the capacitive gap between the electrode(s) and aresonator element. In some embodiments, the mechanical spring constantK_(mechanical) of the accelerometer may be substantially constant duringacceleration-induced motion of the resonator element such that theoverall change in K is caused by changes in K_(electrical).

Some embodiments relate to “static,” resonant, or bulk acoustic waveaccelerometers in which a proof mass is suspended above a substrate by asuspension system. The suspension system may include both z-axissuspension tethers and x-y axis suspension tethers mechanically coupledto the z-axis suspension tethers. The x-y axis suspension tethers may besuspended from the z-axis suspension tethers, and the proof mass may besuspended from the x-y axis suspension tethers.

Some embodiments relate to an accelerometer that includes a resonatorelement configured to vibrate along a first axis in a bulk acoustic wavemode at a resonant frequency. At least one electrode is coupled to theresonator element to excite the resonator element to vibrate along thefirst axis in the bulk acoustic wave mode at the resonant frequency, andto detect a measurement signal.

Some embodiments relate to an accelerometer that includes a resonatorelement having a resonant frequency of at least about 1 MHz. Theresonator element is configured to vibrate along a first axis. At leastone electrode is coupled to the resonator element to excite theresonator element to vibrate along the first axis at the resonantfrequency, and to detect a measurement signal.

Some embodiments relate to a method of measuring motion using anelectromechanical resonator having a resonant frequency and anelectrical spring constant. The electromechanical resonator may includea resonator element. The resonator element is driven to vibrate along afirst axis at the resonant frequency. Motion is measured along the firstaxis based on a change in the electrical spring constant of theelectromechanical resonator.

Some embodiments relate to an accelerometer for measuring motion in anyof a plurality of axes comprising an x-axis, a y-axis, and a z-axis. Theaccelerometer includes a a substrate defining a plane of the x-axis andthe y-axis, and a proof mass suspended above the substrate by asuspension system. The suspension system includes at least one firstsuspension member configured to allow motion of the proof mass along thez-axis. The suspension system also includes at least one secondsuspension member configured to allow motion of the proof mass along thex-axis and/or the y-axis, the at least one second suspension memberbeing mechanically connected to the at least one first suspensionmember. The at least one second suspension member is suspended from theat least one first suspension member.

BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings are not intended to be drawn to scale. In thedrawings, each identical or nearly identical component that isillustrated in various figures is represented by a like referencecharacter. For purposes of clarity, not every component may be labeledin every drawing. In the drawings:

FIG. 1 shows a bulk acoustic wave resonant accelerometer, according tosome embodiments.

FIGS. 2 a-2 b illustrate a bulk acoustic wave mode of vibration of aresonating element.

FIG. 3 shows a scanning electron microscope image of a accelerometerformed in a microfabrication process, according to some embodiments.

FIG. 4 shows a side view of an accelerometer that can measureacceleration in the z-axis direction, according to some embodiments.

FIG. 5 shows a diagram of a circuit for differentially measuringacceleration.

FIG. 6 shows an embodiment of a “static” accelerometer.

FIG. 7 shows a cross sectional view of the accelerometer illustrated inFIG. 6.

FIG. 8 shows another embodiment of a bulk acoustic wave accelerometerthat can operate in a differential mode.

DETAILED DESCRIPTION

The techniques described herein relate to accelerometers and toassociated techniques for detecting and measuring motion. Someembodiments relate to a resonant accelerometer having a resonatingelectromechanical system, in which an externally-applied accelerationcauses a change in the electrical spring constant K_(electrical) of theelectromechanical system. Acceleration can be measured by sensing aresulting change in the resonant frequency of the electromechanicalsystem, and/or a change in another property such as the phase of asignal received from the electromechanical system. In some embodimentsof resonant accelerometers, the accelerometer can be driven to resonatein a bulk acoustic wave (BAW) mode of vibration, which may have a highresonant frequency on the order of 1 MHz to 50 MHz, for example.

Electromechanical systems have both mechanical properties and electricalproperties. One type of electromechanical system is amicroelectromechanical resonator that is electrically driven to vibrateat its resonant frequency. For some microelectromechanical resonators,the mechanical stiffness of the resonator may be modeled as a mechanicalspring constant K_(m), which is a parameter that may be used to modellumped linear systems. A resonator with a large value of K_(m) isrelatively mechanically stiff, and conversely, a resonator with a smallvalue of mechanical spring constant K_(m) is more compliant. Such amicroelectromechanical resonator also has an electrical stiffness thatmay be modeled as an electrical spring constant K_(e), which is anotherparameter that may be used to model lumped linear systems. Theelectrical spring constant K_(e) may represent the strength of theelectrical coupling in the system. Some microelectromechanicalresonators are electrically driven using capacitive coupling, in which avoltage difference between two charged objects, spaced apart by a gap,creates a coulomb force tending to pull the objects together. In anelectromechanical system with capacitive coupling, the electrical springconstant K_(e) may be affected by the size of the gap, the appliedvoltage and/or other parameters. Reducing the size of the gap tends toincrease the strength of the capacitive coupling, which may cause anincrease in the electrical spring constant K_(e).

In some electromechanical systems, the overall stiffness of theelectromechanical system may be represented by an electromechanicalspring constant K_(em) that is a combination of the electrical springconstant K_(e) and the mechanical spring constant K_(m), according tothe relation K_(em)=K_(m)−K_(e). In such an electromechanical system,the resonant frequency of the resonator may be proportional to thesquare root of the electromechanical spring constant K_(em).

As discussed above, prior resonant accelerometers sensed acceleration bydetecting a change in the resonant frequency of the resonator caused bya change in the mechanical spring constant K_(m). These prior resonantaccelerometers were designed so that an externally-applied accelerationwould induce mechanical strain that changed the mechanical springconstant K_(m), which led to a change in the resonant frequency, whichcould be measured to sense acceleration.

Some embodiments of the techniques described herein relate to resonantaccelerometers having an electromechanical resonator that can detectacceleration through a change in the electrical spring constant K_(e).Acceleration and/or other motion parameters can be sensed by measuring achange in the resonant frequency, a phase change in a measured signalreceived from the electromechanical resonator, or a change in anothersuitable parameter that may be linked to the change in the electricalspring constant K_(e). In some embodiments, the mechanical springconstant K_(m) of the system may be substantially fixed or so that itdoes not change, and thus the mechanical spring constant K_(m) may haveno substantial effect on a change in the resonant frequency or otherparameters in response to an externally-applied acceleration.

Some embodiments relate to a capacitive bulk acoustic wave resonantaccelerometer. A capacitive bulk acoustic wave resonant accelerometermay be described as a resonant accelerometer that uses capacitivedriving and/or sensing of a bulk acoustic wave mode of vibration tomeasure acceleration. A bulk acoustic wave (BAW) mode can be describedas a mode of vibration that extends substantially throughout the bulkportion of a resonating element, for example the proof mass or theseismic mass of the accelerometer. The capacitive excitation of a bulkacoustic wave may eliminate the need for a piezoelectric material in thestructure of the accelerometer. This in turn may enable thephotolithographic manufacturing of bulk acoustic wave accelerometers inwidely available substrates such as silicon wafers orsilicon-on-insulator (SOI) wafers. An example of a bulk acoustic wavemode of vibration is shown in FIG. 2 and discussed further below. Arrowsare shown in FIG. 1 to represent the bulk acoustic wave mode ofvibration.

FIG. 1 shows a top view of a bulk acoustic wave resonant accelerometer 1for measuring acceleration along a first axis, according to someembodiments. Accelerometer 1 includes a resonator element 2 that may bedriven to resonate in a bulk acoustic wave mode of vibration. Resonatorelement 2 may be a proof mass that is square or rectangular shape in topview, as shown in FIG. 1, and may have a relatively small height alongthe z-axis direction (not shown in FIG. 1) compared to its width andlength in the x-axis and y-axis directions, respectively. For example,the height of the resonating element may be between 20 μm and 100 μm,and the width and/or length may be on the order of hundreds of microns,such between 300 μm and 800 μm, although other suitable dimensions maybe used. Resonator element 2 may be formed to have other suitable shapesand dimensions, such as a circular shape, as the invention is notlimited to the particular shapes or dimensions of resonator element 2.

Resonating element 2 may be suspended in air (or vacuum, or another gas)by support tethers 3 and 4 such that a space is present below resonatingelement 2. Tethers of any suitable length may be used, and may be in therange of 1 μm, 10 μm, 20 μl, 30 μm, 100 μm or 300 μm in length, and mayhave about the same height as resonator element 2, in some embodiments.FIG. 1 shows that support tethers 3 and 4 are connected to anchors 7 and8. Accelerometer 1 may also include an electrode 5 configured tocapacitively drive resonating element 2 into a bulk acoustic mode ofvibration, and an electrode 6 configured to capacitively sense ameasured signal from the resonating element 2. Resonator element 2 maybe separated from capacitive electrodes 5 and/or 6 by very small gap(s)d, which may be approximately 100 nm to 200 nm in width, for example.For example, gap d may be less than about 200 nm when the resonatorelement is in its equilibrium position and no external acceleration isapplied. Gap d may be along the y-axis direction as illustrated in FIG.1, and for purposes of illustration, FIG. 1 shows the gap d as being inthe y-axis direction. However, the gap d need not necessarily be alongthe y-axis dimension to sense y-axis acceleration. For example, one orboth the resonator element and electrode may have overlapping extensionsthat extend over and/or to the side of the electrode and/or resonator,such that the smallest gap between the resonator element 2 and anelectrode may be their displacement from one another in the x-axisand/or z-axis directions.

Electrode 5 may be connected to an oscillator (not shown) whichgenerates a drive signal that is applied to electrode 5 to excitevibration of a bulk acoustic wave resonance in resonator element 2, atits resonant frequency. The resonant frequency for the bulk acousticmode may be relatively high, and may be between 1 MHz and 50 MHz, suchas about 10 MHz, for example. The mechanical stiffness K_(m) associatedwith this mode may be relatively large. FIGS. 2A and 2B show simulationresults illustrating the vibration of resonator element 2 in a bulkacoustic wave mode. FIG. 2A shows resonating element 2 during a firstportion of a period of the vibration, in which the middle portions ofthe resonator element 2 between the tethers are contracted in the x-yplane towards the center of the resonator element 2. FIG. 2B showsresonating element 2 during a different portion of the period of thevibration, in which the middle portions of the resonator element 2 areexpanded in the x-y plane such that they are bowed away from the centerof the resonator element 2. Such a bulk acoustic wave mode may becharacterized by expansions and contractions of resonator element 2 inthe x-y plane. The center of resonator element 2 and the tethers may notmove as a result of driving the resonator element 2 in the bulk acousticwave mode. In other words, the bulk acoustic resonance mode may notinduce any substantial translation resonator element. The bulk acousticwave modes can exhibit high Quality factors (Q) in the order of 1000 to100,000 or even higher at atmospheric or sub-atmospheric surrounding gaspressure, which may simplify packaging the accelerometer because costlylow-pressure vacuum packaging may not be necessary. In some embodiments,the accelerometer can be packaged in air at atmospheric pressure, orsub-atmospheric pressures of 1 torr or higher.

With respect to FIG. 1, when an external acceleration a_(y) is appliedthat has a component in y-axis direction, the external acceleration maycause the resonator element 2 to be displaced in the y-axis directioncloser to electrode 5. This displacement of resonator element 2 changesthe gap spacing d between the resonator element 2 and the electrodes 5and 6. The resonating element 2 may continue to vibrate in a bulkacoustic wave mode, with the resonating element now being in a positioncloser to electrode 5. This acceleration-induced change in the gapspacing d may cause a change in the electrical spring constant K_(e) ofthe electromechanical system, due to the higher capacitive couplingcaused by reducing the distance d between the resonator element andelectrode 5. As discussed above, such a change in the electrical springconstant K_(e) can cause a change in the resonance frequency of theelectromechanical system and/or other parameters. In some embodiments, achange in resonance frequency may be measured to sense the externalacceleration applied to the accelerometer. However, other ways ofmeasuring the acceleration are possible besides measuring the change inresonance frequency, such as measuring a phase change in a signalreceived from the electromechanical system. Such a phase change may be avery sensitive function of small displacements of the proof mass withrespect to the electrodes due to the high Q of the bulk acousticresonant mode. Any other suitable parameter that reflects or is a resultof the change in K_(e) can be measured to sense the externally-appliedacceleration.

Direct frequency output of the sensor may be connected to digitalcircuitry and measured to a high precision. The resolution with which achange in the output frequency can be measured may depend on the qualityfactor Q of the resonant mode. The use of the bulk acoustic mode of theproof mass may provide for a very high Q (>1000) in air or moderatevacuum, which may result in a high resolution sensor without needinghigh vacuum.

The measurement of the change in resonant frequency, or any othersuitable parameter(s) can be performed in a variety of ways. In someembodiments, a signal can be measured from electrode 6 which is alsoelectrically coupled to resonator element 2. Electrode 6 may detect ameasurement signal that can be processed by electrical circuitry coupledto electrode 6 to detect motion based on a change in the resonantfrequency or another parameter. In some embodiments, the phase shift maybe a more sensitive way to detect motion than detecting a change inresonant frequency. However, another parameter may be used, such asvoltage magnitude, current magnitude, as the invention is not limited asto a particular measurement technique. In some embodiments, ameasurement signal can be detected from the resonator element itself byway of an electrical connections through a tether 3 and/or 4, and ananchor 7 and/or 8, with the anchor or other electrical element coupledto the tether serving as a detection electrode that provides ameasurement signal to measurement circuitry. Such an embodiment isdiscussed in further detail below with respect to FIG. 5.

The components of accelerometer 1 can be formed of any of a variety ofmaterials. In some embodiments, accelerometer 1 may be a micromachinedaccelerometer (e.g., a microelectromechanical system) that is formedusing an integrated circuit manufacturing process, or another suitableprocess for forming microelectromechanical systems (MEMS). Resonatorelement 2 may have release holes formed therein to enable etching of theunderlying material to release the suspended structure. The resonatingelement, tethers, anchors, and/or electrodes may be formed of asemiconductor, such as single crystal silicon or polycrystallinesilicon. However, any other suitable materials may be used, as theinvention is not limited in this respect. One suitable substrate forimplementation of the BAW accelerometer is a silicon-on-insulatorsubstrate, although regular silicon substrate can be used, or anothertype of substrate.

FIG. 3 shows a scanning electron microscope (SEM) image of anaccelerometer 1 that was formed using a micromachining manufacturingprocess, in which the resonating element 2 was formed of single crystalsilicon.

Using accelerometer 1 illustrated in FIG. 1, the component ofacceleration a_(y) along a first axis (e.g., y axis) can be measured. Tomeasure acceleration along a second axis that is perpendicular to thefirst axis (e.g., x axis), a second resonator element can be used withassociated tethers, anchors, and electrode(s) such as those illustratedin FIG. 1, yet oriented in a direction that is shifted by 90° so as todetect acceleration along the x-axis. In some applications, it may bedesirable to use a 3-axis accelerometer that can measure accelerationalong all three axes (x, y and z).

FIG. 4 shows an example of an accelerometer that can sense accelerationin the z-axis direction, according to some embodiments. To senseacceleration in along the z-axis, an electrode can be formed aboveand/or below resonating element 2, and separated from resonating element2 by a gap d. For example, as shown in FIG. 4, a parallel top electrode11 and/or bottom electrode 12 can be used to drive the resonator elementto vibrate in the z-axis direction in a bulk acoustic wave mode ofvibration. Z-axis acceleration can be measured using any suitabletechnique such as those described herein. Electrodes 11 and 12 may bemade from polysilicon or another material, such as a metal. One or moresupporting tethers 3 and 4 can be used with mechanical compliance in thez-axis direction. The tethers 3 and 4 may be made from polysilicon oranother material such as metal (e.g., aluminum or nickel).

FIG. 5 shows an embodiment of a differential bulk acoustic waveaccelerometer and supporting measurement electronics for sensingacceleration along a first axis. In this embodiment, two bulk acousticwave resonator elements 41 and 42 are used in a differentialconfiguration. Electrode 47 is used to capacitively drive resonatorelement 41 into a bulk acoustic wave mode, with electrode 47 beingpositioned on the right side of resonator element 41, as shown in FIG.5. Electrode 48 is used to capacitively drive resonator element 42 intoa bulk acoustic wave mode of vibration, with electrode 48 beingpositioned on the left side of resonator element 42. Acceleration in they-axis direction may cause resonator element 41 to be shifted closer toits drive electrode 47 and resonator element 42 to be shifted fartheraway from its drive electrode 48.

A differential read-out can be used, such that first order common-modevariations caused by temperature can be cancelled out. As illustrated inFIG. 5, each of these resonator elements 41 and 42 may be used in thesingle-electrode configuration, in which each resonator element has anassociated drive electrode, and a sense electrode 6 need not be used. Inthe example illustrated in FIG. 5, drive voltages can be applied to thedrive electrodes, and a sense current may be read out of the body of theresonator via the anchors. Trans-Impedance amplifiers 43 and 44 amplifythe sense current and the two differential output voltages aremultiplied by a mixer 45 and sent through a low-pass filter 46. Theshift in the resonant frequency can be provided as an output, and theacceleration can then be calculated based on this output signal. Oneexample of a measurement configuration is shown in FIG. 5 withassociated circuitry, however this is just one example, and any othersuitable measurement configurations or circuitry may be used.

Acceleration along the y-axis direction may cause the bulk acoustic waveresonant frequency of one of the resonator elements 41 or 42 toincrease, and the resonant frequency of the other one of resonatorelements 41 or 42 to decrease. The increase in the resonant frequency ofone of the resonators may be approximately the same as the decrease inthe resonant frequency of the other resonator (assuming the twocapacitive bulk acoustic resonators operate in their linear range). Thisdifferential mode of operation may result in a two times largerfrequency shift due to the acceleration than would be the case in anon-differential mode of operation. Common-mode frequency driftresulting from a change in the mechanical stiffness or mass of the bulkacoustic wave resonators (for example, caused by temperature) may besubtracted out as a result of the differential mode of operation, suchthat this frequency drift does not appear in the output frequencyF_(out).

FIG. 6 shows an embodiment of another accelerometer 60. Accelerometer 60may be considered to be in the “static” category of accelerometersabove, which are not intentionally driven into a resonance mode likeresonant accelerometers. Accelerometer 60 may be a 3-axis accelerometerthat can sense acceleration in the x, y and z-axes. As shown in FIG. 6,accelerometer 60 has a proof mass 61 that may be suspended above asubstrate by a suspension system that includes x-y axis suspensionmembers and z-axis suspension members. Proof mass 61 may be formed ofsingle-crystal silicon or any other suitable material.

As shown in FIG. 6, proof mass 61 may be suspended by x-y axissuspension tethers 62, and z axis suspensions tethers 66, such that thex-y axis suspension tethers are mechanically anchored to the z-axissuspension tethers. The x-y axis suspension tethers may not exhibit anysubstantial bending in the z-axis acceleration compared to the z-axissuspension tethers. Similarly, the z axis suspension tethers may notexhibit any substantial bending in the x-axis or y-axis compared to x-yaxis suspension tethers. The z-axis suspension tethers may have anL-shape as shown in FIG. 6, or any other suitable shape. In someembodiments, the z-axis suspension tethers may be substantially flat andthin, and may have a thickness of less than about 3 μm. The z-axissuspension tethers may be formed of polysilicon or another suitablematerial. The x-y axis suspension tethers may have a larger height(e.g., greater than 10 μm) than the z-axis suspension tethers in thez-axis direction, so that the x-y suspension tethers do not have anysubstantial compliance in the z-axis direction. As shown in FIG. 6, thex-y suspension tethers may have a folded shape, however, any othersuitable shape may be used. As shown in FIG. 6, an x-y axis suspensiontethers 62 may be connected to a z-axis suspension tether the elbow(i.e., corner) the L-shaped z-axis suspension tether 66.

Acceleration may be capacitively sensed in the x-axis direction and they-axis direction by one or more x-axis electrodes 63 and y-axiselectrodes 64, respectively, which are formed to the side of proof mass61. A z-axis electrode 65 may be formed above proof mass 61 to sensez-axis acceleration. These electrodes may serve as shock stops toprevent the plastic deformation or breakage of the suspensions. Theelectrodes may be displaced from proof mass 61 by very small gaps d,such that the gap size may be less than about 200 nm when the proof massis in its equilibrium position (i.e., when no external acceleration isapplied). The suspensions 62 and 66 may be relatively rigid, so that theproof mass does not move more than about 200 nm.

In some embodiments, accelerometer 60 may be formed on the silicondevice layer of an SOI (silicon-on-insulator) substrate. DRIE (deepreactive ion etching) of the silicon device layer of the SOI wafer maybe used to form the accelerometer, combined with polysilicon surfacemicromachining to form the electrodes 63, 64, and 65. The small gap dfor the x, y or z electrodes may be formed by a thin layer of asacrificial material such as silicon dioxide that is removed at a laterstage of the process flow. Proof mass 60 may be formed by backside DRIE,in some embodiments. The z-axis sense electrode 65 and suspensions 62may be formed from doped polysilicon by surface micromachining.

FIG. 7 shows a side view of the accelerometer 60 along the direction ofarrow B shown in FIG. 6. For clarity, some portions of accelerometer 60are not shown in FIG. 7, such as x-axis electrodes 63. As shown in FIG.7, the proof mass 61 is suspended by a suspension system that includesboth z-axis suspension tethers 66, and x-y suspension tethers 62.Suspension tethers 66 may be rigidly mechanically connected to x-ysuspension tethers 62. A z-axis sense electrode 65 may be formed aboveproof mass 61, and may be separated from proof mass 61 by a very smallgap d. In FIG. 7, sets of arrows illustrate the movement of the z-axissuspension tethers in the z-axis direction, and the x-y axis suspensiontethers in the y-axis direction and the x-axis direction.

FIG. 8 shows an embodiment of another bulk acoustic wave accelerometer80, according to some embodiments. Accelerometer 80 may operate as a3-axis bulk acoustic wave accelerometer system in a differential mode ofoperation. One or more bulk acoustic waves can be excited in x, y and/orz directions by capacitive electrode(s) in the proof masses 71 and 72 ofthe accelerometer 80. The two proof masses 71 and 72 may be mirrorcopies of each other and may be separated by a vertical trench 77 tocreate a differential multi-axis bulk acoustic wave accelerometer systemthat may operate differentially in similar manner the system illustratedin FIG. 5. Sensing may be performed by way of separate sense electrodesuch as sense electrode 6, or using a signal detected from the resonatorelement itself, as in the embodiment of FIG. 5. The width of the trench77 may be less than a micrometer or a few micrometers (for example, 2 or3 micrometers). The suspension tethers shown in FIG. 8 are exemplary, asa three-axis bulk acoustic wave accelerometer invention is not limitedto the particular shapes or dimensions of the suspension tethers. Insome embodiments, additional electrode(s) may be created underneath theproof mass(es) to enable a differential operation of the bulk-acousticwave accelerometer in the z direction. Such electrode may be similar inshape and position to electrode 65.

Although resonator elements 61, 71 and 72 are shown in FIGS. 6-8 ashaving a square or rectangular shape, the resonator elements are notlimited to having a square or rectangular shape. Alternatively, theresonator may have a curved, circular or disk-like shape, in someembodiments. The shapes and/or positions of the electrodes and otherfeatures may be adapted to the shape of the resonator.

Discussion of BAW Operation

The theory of operation of accelerometer 1 will now be discussed infurther detail. A relationship for the sensitivity of the accelerometercan be developed by first considering the change in the gap spacing ddue to a transverse acceleration in the y direction. This accelerationis inversely proportional to the flexural stiffness of theaccelerometer, which is affected by the dimensions of the supporttethers 3, 4.

$\frac{\partial f_{electromechanical}}{\partial a} = {\frac{\partial f_{electromechanical}}{\partial d}\frac{\partial d}{\partial a}}$

An externally applied acceleration (a_(y)) causes the resonator elementto shift closer to the electrode 5 and decreases the gap spacing, d, onone side of resonating element 2, while increasing the gap spacing forelectrode 6 on the opposite side of resonating element 2. The supporttethers each have an associated stiffness given by k₁ and k₂. If the twotethers are have the same dimensions and are formed of the samematerial, then k₁=k₂. The proof mass can be modeled as a 2^(nd)-ordermass spring damper system with spring constant, K_(EFF), damping, D, andmass, M. The two support tethers 3, 4 have dimensions given by l_(t),w_(t), and h_(t). Due to the direction of motion of the proof mass, thetransverse stiffness of the tethers is given by:

$K_{flexure} = {{2\; K_{tethers}} = \frac{2\;{Ew}_{tether}^{3}h_{tether}}{L_{tether}^{3}}}$The proof mass motion can be modeled using a second ordermass-damper-spring system model.

Using F=ma, the external force applied to the resonating element is:

$F_{{{ext}\_{on}}{\_{mass}}} = {{M_{proofmass}\frac{\mathbb{d}^{2}y_{2}}{\mathbb{d}t^{2}}} = {M_{proofmass}\frac{\mathbb{d}^{2}\left( {y + y_{1}} \right)}{\mathbb{d}t^{2}}}}$

The spring force applied by the tethers:

$F_{spring} = {{K_{flexure}y} + {D\frac{\mathbb{d}y}{\mathbb{d}t}}}$

The two forces balance each other

F_(spring) = −F_(ext_on_mass)${{K_{flexure}y} + {D\frac{\mathbb{d}y}{\mathbb{d}t}}} = {{- M_{proofmass}}\frac{\mathbb{d}^{2}\left( {y + y_{1}} \right)}{\mathbb{d}t^{2}}}$${{M_{proofmass}\frac{\mathbb{d}^{2}y}{\mathbb{d}t^{2}}} + {D\frac{\mathbb{d}y}{\mathbb{d}t}} + {K_{flexure}y}} = {{- M_{proofmass}}\frac{\mathbb{d}^{2}y_{1}}{\mathbb{d}t^{2}}}$

The acceleration to be measured is denoted by ‘a’:

$F_{ext} = {{{M_{proofmass}\frac{\mathbb{d}^{2}y}{\mathbb{d}t^{2}}} + {D\frac{\mathbb{d}y}{\mathbb{d}t}} + {K_{flexure}y}} = {aM}_{proofmass}}$

The differential equation in terms of the acceleration to be measured:

$a = {\frac{\mathbb{d}^{2}y}{\mathbb{d}t^{2}} + {\frac{D}{M_{proofmass}}\frac{\mathbb{d}y}{\mathbb{d}t}} + {\frac{K_{flexure}}{M_{proofmass}}y}}$

Taking the Laplace Transform to get the transfer function for thedisplacement vs. the externally applied acceleration:

$\begin{matrix}{{H\left( {j\;\omega} \right)} = \frac{Y\left( {j\;\omega} \right)}{A\left( {j\;\omega} \right)}} \\{= \frac{1}{\left( {j\;\omega} \right)^{2} + {\frac{D}{M_{proofmass}}\left( {j\;\omega} \right)} + \frac{K_{flexure}}{M_{proofmass}}}} \\{= \frac{1}{\left( {\frac{K_{flexure}}{M_{proofmass}} - \omega^{2}} \right) + {j\left( {\frac{D}{M_{proofmass}}\omega} \right)}}}\end{matrix}$

Static devices may operate well below the resonant frequency and thetransfer function simplifies to:

For  ω<< ω_(n):${{H\left( {j\;\omega} \right)}} = {\frac{M_{proofmass}}{K_{flexure}} = \frac{1}{\omega_{flexure}^{2}}}$

The change in the resonant frequency, f_(electromechanical), of theresonator element in Bulk Acoustic Mode can be calculated using a simpleresonator model. Using a simple resonator model, the resonance frequencyfor the width-extensional mode for a homogeneous BAW resonator can bederived as follows. The simple resonator model that is developed for theinfinite plate or long thin rod can be used to model the HarmonicMechanical Resonance Spectrum. The frequency at mode n is given by:

$f_{n} = {\frac{n}{21}\sqrt{\frac{c}{\rho}}}$f_(n) is the natural bulk acoustic frequency, where n is the harmonicnumber, l is the frequency determining dimension, c is the effectiveelastic stiffness constant. For the case of a plate, the shear stiffnessconstant in thickness direction is used, while for the case of the thinrod, Young's Modulus is used. Finally, ρ is the mass density of theresonator material.

For the case of the BAW resonator, the width is the resonancedetermining dimension and thus, the BAW fundamental resonance frequencyis given by the following relationship.

$f_{BAW} = {\frac{1}{2\; W}\sqrt{\frac{E}{\rho}}}$

The relationship for the BAW stiffness constant can now be derived usingthe following relationship between the resonant frequency and the massof the system.

$\omega_{BAW} = \sqrt{\frac{K_{BAW}}{M_{proofmass}}}$

The resulting BAW stiffness is given by

$K_{BAW} = \frac{\pi^{2}{Elh}}{w}$

When an electrostatic force is exerted on the device via a capacitivetransducer, it can be modeled by a second order system using thebulk-acoustic wave stiffness constant

$F_{electrostatic} = {{M_{proofmass}\frac{\mathbb{d}^{2}y}{\mathbb{d}t^{2}}} + {D\frac{\mathbb{d}y}{\mathbb{d}t}} + {K_{BAW}y}}$

The energy stored in a capacitor is given by:

$E_{capacitator} = {\frac{1}{2}{CV}^{2}}$

The electrostatic force is related to the energy stored in the capacitorgiven by

$F_{elecrostatic} = {\frac{\partial E}{\partial y} = {\frac{1}{2}V^{2}\frac{\partial C}{\partial y}}}$

The voltage to be applied consists of a constant polarization (d.c.)voltage and a drive (a.c.) voltage.V _(applied) =V _(dc) +V _(ac)

Let Co denote the rest capacitance of the resonator

$C_{o} = \frac{ɛ\; A}{d}$

Then C, the capacitance of the resonator, is as follows:

$C = {\frac{ɛ\; A}{d - y} = \frac{C_{o}}{1 - \frac{y}{d}}}$

The derivative of the capacitance with respect to distance can beapproximated by a Taylor series expansion and is given by

$\frac{\partial C}{\partial y} = {\frac{C}{{d\left( {1 - \frac{y}{d}} \right)}^{2}} = {\frac{C_{o}}{d}\left( {1 + {\frac{2}{d}y} + {\frac{3}{d^{2}}y^{2}} + \ldots} \right)}}$

The following approximation is only valid if y/d is <<1

$\frac{\partial C}{\partial y} \approx {\frac{C_{o}}{d}\left( {1 + {\frac{2}{d}y}} \right)}$

By using these relationships, the electrostatic force can be related tothe polarization voltage and the capacitance given by

$F_{electrostatic} = {{\frac{1}{2}\left( {V_{p} + V_{ac}} \right)^{2}\frac{\partial C}{\partial y}} = {\frac{1}{2}\left( {V_{p}^{2} + {2\; V_{p}V_{ac}} + V_{ac}^{2}} \right)\frac{C_{o}}{d}\left( {1 + {\frac{2}{d}y}} \right)}}$$F_{electrostatic} \approx {{\frac{1}{2}V_{p}^{2}\frac{C_{o}}{d}} + {V_{p}V_{ac}\frac{C_{o}}{d}} + {V_{p}^{2}\frac{C_{o}}{d^{2}}y}}$

By substituting the electrostatic force into the second ordermass-spring damper system, the following relationship is derived, givenby

${{M_{proofmass}\frac{\mathbb{d}^{2}y}{\mathbb{d}t^{2}}} + {D\frac{\mathbb{d}y}{\mathbb{d}t}} + {\left( {K_{BAW} - {V_{p}^{2}\frac{C_{o}}{d^{2}}}} \right)y}} = {{\frac{1}{2}V_{p}^{2}\frac{C_{o}}{d}} + {V_{p}V_{ac}\frac{C_{o}}{d}}}$

The electromechanical stiffness is a combination of the bulk acousticwave stiffness and the electrical stiffness;K_(electromechanical)=K_(BAW)−K_(electrical), where an expression forthe electrical stiffness is given by

$K_{electrical} = {V_{p}^{2}\frac{C_{o}}{d^{2}}}$

Using these relations, the overall transfer function of displacement vs.applied drive voltage is given by

${{M_{proofmass}s^{2}{Y(s)}} + {{DsY}(s)} + {\left( {K_{BAW} - {V_{p}^{2}\frac{C_{o}}{d^{2}}}} \right){Y(s)}}} = {V_{p}\frac{C_{o}}{d}{V(s)}}$${H(s)} = {\frac{Y(s)}{V(s)} = \frac{V_{p}\frac{C_{o}}{d}}{{M_{proofmass}s^{2}} + {Ds} + \left( {K_{BAW} - {V_{p}^{2}\frac{C_{o}}{d^{2}}}} \right)}}$

The overall electromechanical resonance frequency depends on the appliedvoltage by

$\begin{matrix}{\omega_{electromechanical} = \sqrt{\frac{K_{electromechanical}}{M_{proofmass}}}} \\{= \sqrt{\frac{K_{BAW} - {V_{p}^{2}\frac{C_{o}}{d^{2}}}}{M_{proofmass}}}} \\{= {\omega_{BAW}\sqrt{1 - \frac{V_{p}^{2}C_{o}}{K_{BAW}d^{2}}}}}\end{matrix}$$f_{electromechanical} = {{f_{BAW}\sqrt{1 - \frac{V_{p}^{2}C_{o}}{K_{BAW}d^{2}}}} = {f_{BAW}\sqrt{1 - \frac{V_{p}^{2}ɛ\; A}{K_{BAW}d^{3}}}}}$

By taking the derivative of the electromechanical resonance frequencywith respect to the gap spacing and combining with the flexural modetransfer function, the overall sensitivity can be derived and is givenby

Taylor Series Expansion on

$\frac{1}{\sqrt{1 - \frac{V_{p}^{2}ɛ\; A}{K_{BAW}d^{3}}}}$to simplify the expression:

${f(x)} = \frac{1}{\sqrt{1 - x}}$${p(x)} = {\sum\limits_{n = 0}^{\infty}{\frac{f^{(n)}(0)}{n!}x^{n}}}$${f^{\prime}(x)} = {{\frac{\frac{1}{2}}{\left( {1 - x} \right)^{\frac{3}{2}}}{f^{\prime}(0)}} = \frac{1}{2}}$${f^{''}(x)} = {{\frac{- \frac{3}{4}}{\left( {1 - x} \right)^{\frac{5}{2}}}{f^{''}(0)}} = \frac{- 3}{4}}$${f^{\prime}(x)} = {{\frac{\frac{15}{8}}{\left( {1 - x} \right)^{\frac{7}{2}}}{f^{\prime\prime\prime}(0)}} = {{\frac{15}{8}{p(x)}} \approx {1 + \frac{x}{2}}}}$

Using only up until the 1st order terms:

$\frac{1}{\sqrt{1 - \frac{V_{p}^{2}ɛ\; A}{K_{BAW}d^{3}}}} \approx {1 + \frac{V_{p}^{2}ɛ\; A}{2\; K_{BAW}d^{3}}}$

Substituting:

$\frac{\partial f_{electromechanical}}{\partial d} = {f_{BAW}\frac{1}{2}\left( {1 + \frac{V_{p}^{2}ɛ\; A}{2\; K_{BAW}d^{3}}} \right)\frac{3\; V_{p}^{2}ɛ\; A}{K_{BAW}d^{4}}}$

Finally, the overall sensitivity of the device may be given by thefollowing relationship:

$\frac{\partial f_{electromechanical}}{\partial a} = {\frac{1}{\omega_{flexure}^{2}}\frac{f_{BAW}}{2}\left( {1 + \frac{V_{p}^{2}ɛ\; A}{2\; K_{BAW}d^{3}}} \right)\frac{3\; V_{p}^{2}ɛ\; A}{K_{BAW}d^{4}}}$

Since the sensitivity of the accelerometer can be proportional to thefourth order of the gap spacing, this technique can enable very highsensitivity.

Having now described some illustrative embodiments of the invention, itshould be apparent to those skilled in the art that the foregoing ismerely illustrative and not limiting, having been presented by way ofexample only. Numerous modifications and other illustrative embodimentsmay be contemplated by those of ordinary skill in the art and arebelieved to fall within the scope of the invention. In particular, manyof the examples presented herein involve specific combinations of methodacts or system elements, it should be understood that those acts andthose elements may be combined in other ways to accomplish the sameobjectives. Acts, elements and features discussed only in connectionwith one embodiment are not intended to be excluded from a similar rolein other embodiments.

Use of ordinal terms such as “first,” “second,” “third,” etc. in theclaims to modify a claim element or item in the specification does notby itself connote any priority, presence or order of one element overanother or the temporal order in which acts of a method are performed.Rather, these ordinal terms are used merely as labels to distinguish oneelement having a certain name from another element having a same name,but for the use of the ordinal term, to distinguish the elements.

What is claimed is:
 1. An accelerometer, comprising: a resonator elementconfigured to vibrate along a first axis in a bulk acoustic wave mode ata resonant frequency, the resonator element being suspended in air by asuspension member and having a mechanical spring constant that is asubstantially fixed and a variable electrical spring constant thatdepends on a size of a capacitive gap between the resonator element andat least one electrode, a polarization voltage being applied between theresonator element and the at least one electrode; the suspension memberhaving a dimension substantially different than that of the resonatorelement; the at least one electrode capacitively coupled to theresonator element configured to sense a change in a parameter value tiedto a change in the electrical spring constant and to provide an outputsignal therefor; and an acceleration determination module configured todetermine an amount of external acceleration applied to theaccelerometer based on the output signal provided by the at least oneelectrode.
 2. The accelerometer of claim 1, wherein the parameter valuetied to a change in the electrical spring constant comprises a change inthe resonant frequency of the resonator element, the change in theresonant frequency being caused by a change in the electrical springconstant.
 3. The accelerometer of claim 1, wherein the at least oneelectrode comprises a drive electrode displaced from the resonatorelement by less than 200 nm.
 4. The accelerometer of claim 1, whereinthe change in a parameter value is a change in the resonant frequency ofthe resonator element.
 5. The accelerometer of claim 1, wherein thechange in a parameter value is a phase shift in the resonant frequencyof the resonator element.
 6. The accelerometer of claim 1, wherein theresonance mode is a bulk acoustic wave mode.
 7. The accelerometer ofclaim 1, wherein the resonator element having a resonant frequency of atleast about 1 MHz.
 8. An accelerometer, comprising: a resonator elementhaving a resonant frequency of at least about 1 MHz, the resonatorelement being configured to vibrate along a first axis, the resonatorelement being suspended in air by a suspension member and having avariable electrical spring constant that depends on a size of acapacitive gap between the resonator element and a first electrode, apolarization voltage being applied between the resonator element and thefirst electrode; the suspension member having a dimension substantiallydifferent than that of the resonator element; the first electrodecapacitively coupled to the resonator element configured to sense achange in a parameter value tied to a change in the electrical springconstant and provide an output signal therefor; and an accelerationdetermination module configured to determine an amount of externalacceleration applied to the accelerometer based on the output signalprovided by the first electrode.
 9. The accelerometer of claim 8,wherein the parameter value tied to a change in the electrical springconstant comprises the resonant frequency being caused by a change inthe electrical spring constant.
 10. The accelerometer of claim 8,wherein the resonator element has a mechanical spring constant that is asubstantially fixed.
 11. The accelerometer of claim 8, wherein the firstelectrode is displaced from the mechanical resonator element by lessthan about 200 nm.
 12. The accelerometer of claim 8, having a qualityfactor of at least 1,000.
 13. An accelerometer, comprising: a resonatorelement comprising a proof mass suspended relative to a substrate by atleast one suspension member attached thereto and configured to allowmotion of the proof mass, the proof mass configured to vibrate along afirst axis in a bulk acoustic resonance mode at a resonant frequency andthe suspension member configured to vibrate in a flexural mode at afrequency different than the resonant frequency, the resonator elementhaving a variable electrical spring constant that depends on a size of acapacitive gap between the resonator element and a first electrode, apolarization voltage being applied between the resonator element and thefirst electrode; and a parameter sensing electrode configured to sense achange in a parameter value tied to a change in the electrical springconstant and provide an acceleration determination signal.
 14. Theaccelerometer of claim 13 wherein the resonant frequency is greater thanthe frequency at which the suspension member vibrates.
 15. Theaccelerometer of claim 13 wherein the resonant frequency is an order ofmagnitude greater than the frequency at which the suspension membervibrates.
 16. An accelerometer, comprising: a drive electrode; aresonator element comprising a proof mass suspended relative to asubstrate by at least one suspension member attached thereto andconfigured to allow the proof mass to be separated from the driveelectrode by a capacitive gap and driven by the drive electrode tovibrate along a first axis in a bulk acoustic resonance mode at aresonant frequency, the suspension member having dimensionssubstantially different than the proof mass and configured to vibrate ina flexural mode, the resonator element having a variable electricalspring constant that depends on a size of a capacitive gap between theresonator element and a first electrode, a polarization voltage beingapplied between the resonator element and the first electrode; and aparameter sensing electrode configured to sense a change in a parametervalue tied to a change in the electrical spring constant and provide anacceleration determination signal.
 17. The accelerometer of claim 16,wherein the change in a parameter value is a change in the resonantfrequency of the resonator element.
 18. The accelerometer of claim 16,wherein the change in a parameter value is a phase shift in the resonantfrequency of the resonator element.
 19. The accelerometer of claim 16wherein the resonator element comprises a pair of suspension members.